ITP001 Axioms: ITP142^7.ax
%------------------------------------------------------------------------------
% File : ITP142^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : util_prob.ax [Gau19]
% : HL4142^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 150 ( 26 unt; 66 typ; 0 def)
% Number of atoms : 316 ( 52 equ; 16 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 1180 ( 16 ~; 2 |; 48 &;1039 @)
% ( 17 <=>; 58 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg;1039 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 240 ( 240 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 63 usr; 4 con; 0-6 aty)
% Number of variables : 264 ( 19 ^ 176 !; 29 ?; 264 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eutil__prob_2Edisjoint,type,
c_2Eutil__prob_2Edisjoint:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Etransc_2Eexp,type,
c_2Etransc_2Eexp: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Einf,type,
c_2Ereal_2Einf: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eutil__prob_2Elg,type,
c_2Eutil__prob_2Elg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Eln,type,
c_2Etransc_2Eln: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eutil__prob_2Elogr,type,
c_2Eutil__prob_2Elogr: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eutil__prob_2Eminimal,type,
c_2Eutil__prob_2Eminimal: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Eutil__prob_2Epair,type,
c_2Eutil__prob_2Epair:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Eutil__prob_2Epowr,type,
c_2Eutil__prob_2Epowr: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eutil__prob_2Eprod__sets,type,
c_2Eutil__prob_2Eprod__sets:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( A_27a > $o ) > $o ) > ( ( A_27b > $o ) > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eutil__prob_2Epair__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > $o,V1Y: A_27b > $o] :
( ( c_2Eutil__prob_2Epair @ A_27a @ A_27b @ V0X @ V1Y )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V2x: A_27a,V3y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0X ) @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V1Y ) ) ) ) ).
thf(thm_2Eutil__prob_2Epowr__def,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Eutil__prob_2Epowr @ V0x @ V1a )
= ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__mul @ V1a @ ( c_2Etransc_2Eln @ V0x ) ) ) ) ).
thf(thm_2Eutil__prob_2Elogr__def,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Eutil__prob_2Elogr @ V0a @ V1x )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eln @ V1x ) @ ( c_2Etransc_2Eln @ V0a ) ) ) ).
thf(thm_2Eutil__prob_2Elg__def,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Eutil__prob_2Elg @ V0x )
= ( c_2Eutil__prob_2Elogr @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ).
thf(thm_2Eutil__prob_2Eprod__sets__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: ( A_27a > $o ) > $o,V1b: ( A_27b > $o ) > $o] :
( ( c_2Eutil__prob_2Eprod__sets @ A_27a @ A_27b @ V0a @ V1b )
= ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27b > $o ) )
@ ( c_2Epair_2EUNCURRY @ ( A_27a > $o ) @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ $o )
@ ^ [V2s: A_27a > $o,V3t: A_27b > $o] : ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ $o @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V2s @ V3t ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V0a ) @ ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3t @ V1b ) ) ) ) ) ) ).
thf(thm_2Eutil__prob_2Eminimal__def,axiom,
! [V0p: tyop_2Enum_2Enum > $o] :
( ( c_2Eutil__prob_2Eminimal @ V0p )
= ( c_2Emin_2E_40 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Ebool_2E_2F_5C @ ( V0p @ V1n )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2m: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Eprim__rec_2E_3C @ V2m @ V1n ) @ ( c_2Ebool_2E_7E @ ( V0p @ V2m ) ) ) ) ) ) ) ).
thf(thm_2Eutil__prob_2Edisjoint__def,axiom,
! [A_27a: $tType,V0A: ( A_27a > $o ) > $o] :
( ( c_2Eutil__prob_2Edisjoint @ A_27a @ V0A )
<=> ! [V1a: A_27a > $o,V2b: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1a @ V0A )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2b @ V0A )
& ( (~) @ ( V1a = V2b ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V1a @ V2b ) ) ) ).
thf(thm_2Eutil__prob_2EEQ__T__IMP,axiom,
! [V0x: $o] :
( V0x
<=> ( c_2Ebool_2ET
=> V0x ) ) ).
thf(thm_2Eutil__prob_2EIN__PAIR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1X: A_27a > $o,V2Y: A_27b > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0x @ ( c_2Eutil__prob_2Epair @ A_27a @ A_27b @ V1X @ V2Y ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x ) @ V1X )
& ( c_2Ebool_2EIN @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x ) @ V2Y ) ) ) ).
thf(thm_2Eutil__prob_2EPAIR__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Eutil__prob_2Epair @ A_27a @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
= ( c_2Epred__set_2EUNIV @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(thm_2Eutil__prob_2EPAIRED__BETA__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1z: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f @ V1z )
= ( V0f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1z ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1z ) ) ) ).
thf(thm_2Eutil__prob_2Elg__1,axiom,
( ( c_2Eutil__prob_2Elg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Eutil__prob_2Elogr__1,axiom,
! [V0b: tyop_2Erealax_2Ereal] :
( ( c_2Eutil__prob_2Elogr @ V0b @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Eutil__prob_2Elg__nonzero,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) )
=> ( ( (~)
@ ( ( c_2Eutil__prob_2Elg @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
<=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eutil__prob_2Elg__mul,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( ( c_2Eutil__prob_2Elg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eutil__prob_2Elg @ V0x ) @ ( c_2Eutil__prob_2Elg @ V1y ) ) ) ) ).
thf(thm_2Eutil__prob_2Elogr__mul,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2y ) )
=> ( ( c_2Eutil__prob_2Elogr @ V0b @ ( c_2Erealax_2Ereal__mul @ V1x @ V2y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eutil__prob_2Elogr @ V0b @ V1x ) @ ( c_2Eutil__prob_2Elogr @ V0b @ V2y ) ) ) ) ).
thf(thm_2Eutil__prob_2Elg__2,axiom,
( ( c_2Eutil__prob_2Elg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eutil__prob_2Elg__inv,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Eutil__prob_2Elg @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Eutil__prob_2Elg @ V0x ) ) ) ) ).
thf(thm_2Eutil__prob_2Elogr__inv,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Eutil__prob_2Elogr @ V0b @ ( c_2Erealax_2Einv @ V1x ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Eutil__prob_2Elogr @ V0b @ V1x ) ) ) ) ).
thf(thm_2Eutil__prob_2Elogr__div,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2y ) )
=> ( ( c_2Eutil__prob_2Elogr @ V0b @ ( c_2Ereal_2E_2F @ V1x @ V2y ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Eutil__prob_2Elogr @ V0b @ V1x ) @ ( c_2Eutil__prob_2Elogr @ V0b @ V2y ) ) ) ) ).
thf(thm_2Eutil__prob_2Eneg__lg,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Erealax_2Ereal__neg @ ( c_2Eutil__prob_2Elg @ V0x ) )
= ( c_2Eutil__prob_2Elg @ ( c_2Erealax_2Einv @ V0x ) ) ) ) ).
thf(thm_2Eutil__prob_2Eneg__logr,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Erealax_2Ereal__neg @ ( c_2Eutil__prob_2Elogr @ V0b @ V1x ) )
= ( c_2Eutil__prob_2Elogr @ V0b @ ( c_2Erealax_2Einv @ V1x ) ) ) ) ).
thf(thm_2Eutil__prob_2Elg__pow,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eutil__prob_2Elg @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ V0n ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ,axiom,
? [V0f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__INV,axiom,
? [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ,axiom,
? [V0f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__INV,axiom,
? [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT,axiom,
? [V0f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT__INV,axiom,
? [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT2,axiom,
? [V0f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT2__INV,axiom,
? [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EIN__o,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1f: A_27a > A_27b,V2s: A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Ecombin_2Eo @ A_27a @ $o @ A_27b @ V2s @ V1f ) )
= ( c_2Ebool_2EIN @ A_27b @ ( V1f @ V0x ) @ V2s ) ) ).
thf(thm_2Eutil__prob_2EIN__PROD__SETS,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1a: ( A_27a > $o ) > $o,V2b: ( A_27b > $o ) > $o] :
( ( c_2Ebool_2EIN @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ V0s @ ( c_2Eutil__prob_2Eprod__sets @ A_27a @ A_27b @ V1a @ V2b ) )
<=> ? [V3t: A_27a > $o,V4u: A_27b > $o] :
( ( V0s
= ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V3t @ V4u ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ V1a )
& ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V4u @ V2b ) ) ) ).
thf(thm_2Eutil__prob_2Efinite__enumeration__of__sets__has__max__non__empty,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ! [V2x: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V0f @ V2x ) @ V1s )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( (~) @ ( V3m = V4n ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0f @ V3m ) @ ( V0f @ V4n ) ) ) )
=> ? [V5N: tyop_2Enum_2Enum] :
! [V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V6n @ V5N )
=> ( ( V0f @ V6n )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL1,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ECOMPL @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V0c @ V1x ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ V2x @ V0c ) ) ) ) ).
thf(thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL2,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ECOMPL @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V0c @ V1x ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ V2x @ V0c ) ) ) ) ).
thf(thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL3,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ECOMPL @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V1x @ V0c ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ V0c @ V2x ) ) ) ) ).
thf(thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL4,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ECOMPL @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V1x @ V0c ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ V0c @ V2x ) ) ) ) ).
thf(thm_2Eutil__prob_2EGBIGUNION__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a,V1p: A_27b > A_27c > $o,V2n: A_27d] :
( ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V3s: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V3s
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V4n: A_27c] : ( V1p @ V3s @ V4n ) ) ) )
= ( c_2Epred__set_2EBIGUNION @ A_27b
@ ( c_2Epred__set_2EIMAGE @ A_27c @ ( A_27b > $o )
@ ^ [V5n: A_27c] :
( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V6s: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V6s @ ( V1p @ V6s @ V5n ) ) )
@ ( c_2Epred__set_2EUNIV @ A_27c ) ) ) ) ).
thf(thm_2Eutil__prob_2EPOW__HALF__POS,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0n ) ) ).
thf(thm_2Eutil__prob_2EPOW__HALF__SMALL,axiom,
! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e )
=> ? [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) @ V0e ) ) ).
thf(thm_2Eutil__prob_2EPOW__HALF__MONO,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0m ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LE__LT__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__LE__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__MUL__IDEMPOT,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0r @ V0r )
= V0r )
<=> ( ( V0r
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V0r
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__SUP__LE__X,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2r: tyop_2Erealax_2Ereal] : ( V0P @ V2r )
& ! [V3r: tyop_2Erealax_2Ereal] :
( ( V0P @ V3r )
=> ( c_2Ereal_2Ereal__lte @ V3r @ V1x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0P ) @ V1x ) ) ).
thf(thm_2Eutil__prob_2EREAL__X__LE__SUP,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2r: tyop_2Erealax_2Ereal] : ( V0P @ V2r )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4r: tyop_2Erealax_2Ereal] :
( ( V0P @ V4r )
=> ( c_2Ereal_2Ereal__lte @ V4r @ V3z ) )
& ? [V5r: tyop_2Erealax_2Ereal] :
( ( V0P @ V5r )
& ( c_2Ereal_2Ereal__lte @ V1x @ V5r ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Esup @ V0P ) ) ) ).
thf(thm_2Eutil__prob_2EINF__DEF__ALT,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal_2Einf @ V0p )
= ( c_2Erealax_2Ereal__neg
@ ( c_2Ereal_2Esup
@ ^ [V1r: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1r ) @ V0p ) ) ) ) ).
thf(thm_2Eutil__prob_2ELE__INF,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1r: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0p )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0p )
=> ( c_2Ereal_2Ereal__lte @ V1r @ V3x ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1r @ ( c_2Ereal_2Einf @ V0p ) ) ) ).
thf(thm_2Eutil__prob_2EINF__LE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1r: tyop_2Erealax_2Ereal] :
( ( ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0p )
=> ( c_2Ereal_2Ereal__lte @ V2z @ V3x ) )
& ? [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0p )
& ( c_2Ereal_2Ereal__lte @ V4x @ V1r ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Einf @ V0p ) @ V1r ) ) ).
thf(thm_2Eutil__prob_2EINF__GREATER,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1z: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0p )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Einf @ V0p ) @ V1z ) )
=> ? [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0p )
& ( c_2Erealax_2Ereal__lt @ V3x @ V1z ) ) ) ).
thf(thm_2Eutil__prob_2EINF__CLOSE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0p )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) )
=> ? [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0p )
& ( c_2Erealax_2Ereal__lt @ V3x @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Einf @ V0p ) @ V1e ) ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__NEG__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__LMUL__0__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__RMUL__0__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
& ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__LMUL__NEG__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) )
=> ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__RMUL__NEG__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__LMUL__NEG__0__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__RMUL__NEG__0__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Eutil__prob_2EREAL__LT__RDIV__EQ__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V1y @ V2z ) @ V0x )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2EREAL__LE__RDIV__EQ__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V1y @ V2z ) @ V0x )
= ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2EPOW__POS__EVEN,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ V1x @ V0n ) )
= ( c_2Earithmetic_2EEVEN @ V0n ) ) ) ).
thf(thm_2Eutil__prob_2EPOW__NEG__ODD,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ V1x @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Earithmetic_2EODD @ V0n ) ) ) ).
thf(thm_2Eutil__prob_2ELOGR__MONO__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2b ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Eutil__prob_2Elogr @ V2b @ V0x ) @ ( c_2Eutil__prob_2Elogr @ V2b @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2ELOGR__MONO__LE__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2b ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eutil__prob_2Elogr @ V2b @ V0x ) @ ( c_2Eutil__prob_2Elogr @ V2b @ V1y ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__EXISTS0,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
<=> ? [V2n: tyop_2Enum_2Enum] :
( ( V0P @ V2n )
& ! [V3m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V2n )
=> ( (~) @ ( V0P @ V3m ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__EXISTS,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
<=> ( ( V0P @ ( c_2Eutil__prob_2Eminimal @ V0P ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Eutil__prob_2Eminimal @ V0P ) )
=> ( (~) @ ( V0P @ V2n ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__EXISTS__IMP,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
=> ? [V2m: tyop_2Enum_2Enum] :
( ( V0P @ V2m )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ V2m )
=> ( (~) @ ( V0P @ V3n ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__EQ__IMP,axiom,
! [V0m: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum > $o] :
( ( ( V1p @ V0m )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V0m )
=> ( (~) @ ( V1p @ V2n ) ) ) )
=> ( V0m
= ( c_2Eutil__prob_2Eminimal @ V1p ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum > $o] :
( ( ( ( c_2Enum_2ESUC @ V0n )
= ( c_2Eutil__prob_2Eminimal @ V1p ) )
& ( V1p @ ( c_2Enum_2ESUC @ V0n ) ) )
<=> ( ( (~) @ ( V1p @ c_2Enum_2E0 ) )
& ( V0n
= ( c_2Eutil__prob_2Eminimal @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ $o @ tyop_2Enum_2Enum @ V1p @ c_2Enum_2ESUC ) ) )
& ( V1p @ ( c_2Enum_2ESUC @ V0n ) ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__EQ,axiom,
! [V0p: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum] :
( ( ( V0p @ V1m )
& ( V1m
= ( c_2Eutil__prob_2Eminimal @ V0p ) ) )
<=> ( ( V0p @ V1m )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1m )
=> ( (~) @ ( V0p @ V2n ) ) ) ) ) ).
thf(thm_2Eutil__prob_2EMINIMAL__SUC__IMP,axiom,
! [V0n: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum > $o] :
( ( ( V1p @ ( c_2Enum_2ESUC @ V0n ) )
& ( (~) @ ( V1p @ c_2Enum_2E0 ) )
& ( V0n
= ( c_2Eutil__prob_2Eminimal @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ $o @ tyop_2Enum_2Enum @ V1p @ c_2Enum_2ESUC ) ) ) )
=> ( ( c_2Enum_2ESUC @ V0n )
= ( c_2Eutil__prob_2Eminimal @ V1p ) ) ) ).
thf(thm_2Eutil__prob_2EdisjointI,axiom,
! [A_27a: $tType,V0A: ( A_27a > $o ) > $o] :
( ! [V1a: A_27a > $o,V2b: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1a @ V0A )
=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2b @ V0A )
=> ( ( (~) @ ( V1a = V2b ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V1a @ V2b ) ) ) )
=> ( c_2Eutil__prob_2Edisjoint @ A_27a @ V0A ) ) ).
thf(thm_2Eutil__prob_2EdisjointD,axiom,
! [A_27a: $tType,V0A: ( A_27a > $o ) > $o,V1a: A_27a > $o,V2b: A_27a > $o] :
( ( c_2Eutil__prob_2Edisjoint @ A_27a @ V0A )
=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1a @ V0A )
=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2b @ V0A )
=> ( ( (~) @ ( V1a = V2b ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V1a @ V2b ) ) ) ) ) ).
thf(thm_2Eutil__prob_2Edisjoint__empty,axiom,
! [A_27a: $tType] : ( c_2Eutil__prob_2Edisjoint @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ).
thf(thm_2Eutil__prob_2Edisjoint__union,axiom,
! [A_27a: $tType,V0A: ( A_27a > $o ) > $o,V1B: ( A_27a > $o ) > $o] :
( ( ( c_2Eutil__prob_2Edisjoint @ A_27a @ V0A )
& ( c_2Eutil__prob_2Edisjoint @ A_27a @ V1B )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0A ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1B ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( c_2Eutil__prob_2Edisjoint @ A_27a @ ( c_2Epred__set_2EUNION @ ( A_27a > $o ) @ V0A @ V1B ) ) ) ).
thf(thm_2Eutil__prob_2Edisjoint__sing,axiom,
! [A_27a: $tType,V0a: A_27a > $o] : ( c_2Eutil__prob_2Edisjoint @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0a @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) ).
%------------------------------------------------------------------------------